> [!tldr] > The Holt-Winters Procedure is a family of time series forecasting procedures. It uses updating equations to model (1) the local level $L_{t}$, (2) the long-term trend $T_{t}$, and (3) the seasonal effects $S_{t}$ of a time series. For non-seasonal time series, the procedure is called Holt's procedure, with the updating equations $\begin{align*} L_{t} &= \alpha x_{t} + (1-\alpha)(L_{t-1}+T_{t-1})\\ T_{t} &= \beta(L_{t}-L_{t-1}) + (1-\beta)T_{t-1} \end{align*}$where $\alpha, \gamma$ are parameters. It gives forecasts $\hat{x}_{N+h}=L_{N}+hT_{N}$. With additive seasonality of period $m$, the Holt-Winters procedure use the updating equations $\begin{align*} L_{t} &= \alpha (x_{t}-S_{t-m}) + (1-\alpha)(L_{t-1}+T_{t-1})\\ T_{t} &= \beta(L_{t}-L_{t-1}) + (1-\beta)T_{t-1}\\ S_{t} &= \gamma(x_{t} - L_{t}) + (1-\gamma)S_{t-m} \end{align*}$and for multiplicative seasonality, replace $x_{t}- \square$ with $x_t / \square$. ### Damped Holt-Winters For more realistic forecasts, damped Holt-Winters (without seasonality for simplicity) scales down the trend by exponentially decaying weights $\phi^{i}$: $\hat{x}_{N+h}=L_{N} + \sum_{i=1}^{h}\phi^{i}T_{N}$